Rule formats for bounded nondeterminism in structural operational - PowerPoint PPT Presentation

Rule formats for bounded nondeterminism in structural operational semantics Álvaro García-Pérez Luca Aceto Anna Ingólfsdóttir Reykjavík University Lyngby, January 8th, 2016 1 / 11

Motivation 2 / 11

Structural operational semantics and bounded nondeterminism A transition system specification (TSS) consists of inference rules that a induce a labelled transition system (LTS) { p − → p ′ } 3 / 11

Structural operational semantics and bounded nondeterminism A transition system specification (TSS) consists of inference rules that a induce a labelled transition system (LTS) { p − → p ′ } Exercises 3.3 and 3.4 in Semantics with Applications: An Appetizer [Nielson and Nielson, 2007] While language with nondeterminisitc choice and statement random( x ) . x:=-1; while x<=0 do (x:=x-1 or x:=(-1)*x) a An LTS is finite branching iff for every p , the set { ( a , p ′ ) | p − → p ′ } is finite. 3 / 11

Structural operational semantics and bounded nondeterminism A transition system specification (TSS) consists of inference rules that a induce a labelled transition system (LTS) { p − → p ′ } Exercises 3.3 and 3.4 in Semantics with Applications: An Appetizer [Nielson and Nielson, 2007] While language with nondeterminisitc choice and statement random( x ) . x:=-1; while x<=0 do (x:=x-1 or x:=(-1)*x) a An LTS is finite branching iff for every p , the set { ( a , p ′ ) | p − → p ′ } is finite. Rule formats for finite branching: statically checkable (ideally) conditions on TSSs that guarantee continuous Scott-Strachey semantics ([Apt and Plotkin, 1986]). 3 / 11

Existing rule format for finite branching [Fokkink and Vu, 2003] Theorem (Correctness of rule format) Let R be a TSS. The LTS associated to R is finite branching if the following conditions hold: (i) R has no unguarded recursion ( strict stratification ). (ii) Each rule in R gives rise to finitely many transitions from each process ( bounded nondeterminism format ). (iii) Only finitely many rules in R can give rise to transitions from each process ( uniformity and finitely inhabited η -types ). 4 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 5 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 Strict stratification: η S ( c ) = 0 S ( p 0 � p 1 ) = 1 + S ( p 0 ) + S ( p 1 ) . . . 5 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 Bounded nondeterminism format: η b k � � u k u ′ k a t t ′ 5 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 Uniformity and finitely inhabited η -types: η ( x 0 � x 1 ) = { x 0 , x 1 } 5 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 Uniformity and finitely inhabited η -types: � x 0 � x 1 , { x 0 �→ { c } , x 1 �→ ∅}� η ( x 0 � x 1 ) = { x 0 , x 1 } 5 / 11

Example (Rules for merge in BPA) c c → x ′ → x ′ x 0 − x 1 − 0 1 . . . c c . . . → x ′ → x 0 � x ′ x 0 � x 1 − 0 � x 1 x 0 � x 1 − 1 Uniformity and finitely inhabited η -types: � x 0 � x 1 , { x 0 �→ { c } , x 1 �→ ∅}� � x 0 � x 1 , { x 0 �→ ∅ , x 1 �→ { c }}� η ( x 0 � x 1 ) = { x 0 , x 1 } 5 / 11

The problem ◮ Mechanising the proof of correctness of the rule format? Claim [Fokkink and Vu, 2003] For every term t there are finitely many maps ψ such that there exists a rule r of η -type � t , ψ � which gives rise to transitions. Proof: by assuming that the set of different maps ψ is infinite and deriving a contradiction. Reasoning by contradiction here is not constructive! ◮ Bounded-nondeterminism properties other than finite branching? An LTS is image finite iff for every p and a the set { p ′ | p a − → p ′ } is finite. a An LTS is initials finite iff for every p the set { a | ∃ p ′ . p − → p ′ } is finite. Rule formats for initials finiteness and for finite branching? 6 / 11

Our contribution 7 / 11

Constructive proof of correcteness of the rule format For each process p = σ ( t ) , the ψ maps such that there exists a rule r of η -type � t , ψ � which gives rise to transitions are dependent functions of a type ψ : Π v ∈ η ( t ) { a | σ ( v ) − → q } . Constructivity enables the mechanisation of the proof with a state-of-the-art proof assistant (work in progress). 8 / 11

Rule formats for image finiteness and initials finiteness Definition (Image finiteness and initials finiteness) An LTS is image finite iff for every p and a the set { p ′ | p a − → p ′ } is finite. a An LTS is initials finite iff for every p the set { a | ∃ p ′ . p − → p ′ } is finite. The properties require modified η -types that either ignore the targets or keep track of both actions and targets in transitions. 9 / 11

Rule formats for image finiteness and initials finiteness Definition (Image finiteness and initials finiteness) An LTS is image finite iff for every p and a the set { p ′ | p a − → p ′ } is finite. a An LTS is initials finite iff for every p the set { a | ∃ p ′ . p − → p ′ } is finite. The properties require modified η -types that either ignore the targets or keep track of both actions and targets in transitions. Example (Statement random( x ) ) n ∈ N . , n � random( x ) ; S , s � − → � S , s [ x �→ n ] � 9 / 11

Related and Future work ◮ Generalise the rule formats to other bounded-nondeterminism properties [Aceto et al., 2016]. ◮ Extend the rule formats to SOS with terms as labels [Aceto et al., 2016]. ◮ Modify the rule formats to cover cases that we are aware are not covered yet. ◮ Extend the rule formats to many sorted signatures and Nominal SOS. 10 / 11

Summary ◮ Rule formats for bounded nondeterminism are useful to check whether a language admits a standard continuous semantics a la Scott-Strachey. ◮ We provide a constructive proof of correctness of the rule format for finite branching in [Fokkink and Vu, 2003]. ◮ We provide rule formats for initials finiteness and image finiteness. 11 / 11

Summary ◮ Rule formats for bounded nondeterminism are useful to check whether a language admits a standard continuous semantics a la Scott-Strachey. ◮ We provide a constructive proof of correctness of the rule format for finite branching in [Fokkink and Vu, 2003]. ◮ We provide rule formats for initials finiteness and image finiteness. Happy Birthday to Hanne and Flemming! 11 / 11

References I Aceto, L., Fábregas, I., García-Pérez, A., and Ingólfsdóttir, A. (2016). A unified rule format for bounded nondeterminism in SOS with terms as labels. Submitted. Apt, K. R. and Plotkin, G. D. (1986). Countable nondeterminism and random assignment. Journal of the ACM , 33(4):724–767. Fokkink, W. and Vu, T. D. (2003). Structural operational semantics and bounded nondeterminism. Acta Informatica , 39(6-7):501–516. Nielson, H. R. and Nielson, F. (2007). Semantics with Applications: An Appetizer . Springer-Verlag New York. 11 / 11